329 ten thousandths<br> is how much greater than 32 thousandths

dmitriy555 [2]

3x²cos( x³ ) and 3sin²( x ) cos( x ) are the derivatives of the composite functions f(x) = sin(x³) and f(x) = sin³(x) respectively.

<h3>What are the derivative of f(x) = sin(x³) and f(x) = sin³(x)?</h3>

Chain rule simply shows how to find the derivative of a composite function. It states that;

d/dx[f(g(x))] = f'(g(x))g'(x)

Given the data in the question;

f(x) = sin(x³) = ?

f(x) = sin³(x) = ?

First, we find the derivate of the composite function f(x) = sin(x³) using chain rule.

d/dx[f(g(x))] = f'(g(x))g'(x)

f(x) = sin(x)

g(x) = x³

Apply chain rule, set u as x³

d/du[ sin( u )] d/dx[ x³ ]

cos( u ) d/dx[ x³ ]

cos( x³ ) d/dx[ x³ ]

Now, differentiate using power rule.

d/dx[ xⁿ ] is nxⁿ⁻¹

cos( x³ ) d/dx[ x³ ]

In our case, n = 3

cos( x³ ) ( 3x² )

Reorder the factors

3x²cos( x³ )

Next, we find the derivative of f(x) = sin³(x)

d/dx[f(g(x))] = f'(g(x))g'(x)

f( x ) = x³

g( x ) = sin( x )

Apply chain rule, set u as sin( x )

d/du[ u³ ] d/dx[ sin( x )]

Now, differentiate using power rule.

d/dx[ xⁿ ] is nxⁿ⁻¹

d/du[ u³ ] d/dx[ sin( x )]

3u² d/dx[ sin( x )]

Replace the u with sin( x )

3sin²(x) d/dx[ sin( x )]
Derivative of sin x with respect to x is cos (x)

3sin²( x ) cos( x )

Therefore, the derivatives of the functions are 3x²cos( x³ ) and 3sin²( x ) cos( x ).

Learn more about chain rule here: brainly.com/question/2285262

#SPJ1

4 0

1 year ago

At a certain vineyard it is found that each grape vine produces about 10 lb of grapes in a season when about 700 vines are plant

Delicious77 [7]

Answer:

Number of vines that should be planted are 150.

Step-by-step explanation:

The number of pounds of pounds of grapes produced per acre is represented by the expression $A_{n}=(700+n)(10-0.01n)$

Where n = additional vines planted

To maximize the production of grapes we will find the derivative of A(n) and equate it to zero.

$A_{n}=(7000+3n-0.01n^{2} )$

$A'_{n}=(3-0.02n)$

For $A'_{n}=0$

3 - 0.02n = 0

0.02n = 3

n = $\frac{3}{0.02}$

n = 150

To check whether the maximum value of the function is at n = 150, we will find the second derivative A(n).

$A''_{n}=-0.02$

Which shows A"(n) < 0

Therefore, A(n) has the maximum value at n = 150.

Therefore, number of vines that should be planted are 150.

5 0

4 years ago

Someone please help me?!?!

VikaD [51]

Answer:

30 is correct answer. ... ...........

5 0

3 years ago

What are the domain and range of f(x) = 2x - 4?

Anni [7]

Answer:

y intercept is -4 and the slope is 2. You can figure it out from there. Just make a graph

Step-by-step explanation:

7 0

3 years ago

Read 2 more answers

Simplify: -4.6m + 1.8m

ra1l [238]

-4.6+1.8 is -2.8
You have to do simple math

6 0

3 years ago

Read 2 more answers

329 ten thousandths is how much greater than 32 thousandths